I see the saves in the directory, but the game doesn't seem to see them. I can only start a new game. I can't load a saved game to resume it. All slots show up as empty, in spite of the files being there. What am I missing?

Have you read the question?
He never mentioned save files with strange names.
Anyway,I have the same issue.I have 3 save files in the Save directory,but Rune doesn't see them.
From what I get,saving the game from the "Save Game" option in the main menu allows them to be loaded without issues. Quicksaves (Rune doesn't see them) and Autosaves (the "Load Last Autosave" button only leads to a black screen) don't seem to work.

Okay,I think I get it now.Apparently,Quicksaves only work ingame.That means that they will be wiped out after you close the game.As for autosaves,they glitched once for me but they are working now.
So I suggest saving manually from time to time and before closing the game.

I will give it another try. Thanks for looking into it. I get the feeling it might have to do with the different location of "My Documents" in Windows 7. Has anyone tried installing Gold instead of Classic? Has it got the same issue?

I reinstalled, however, I think the issue was that I might have tried to save too soon, before the credits. I tried a save after the shipwreck and it worked fine, so I'm thinking if you try to save during the intro it creates bad saves that crash the game. Maybe someone can confirm? If you are having issues saving, try playing past the shipwreck.

Thanks!